Post on 23-Jul-2020
Thesis Defense:Adaptive Binary Search Trees
Jonathan C. Derryberry
Department of Computer ScienceCarnegie Mellon University
December 16, 2009
Thesis Committee:
Daniel Sleator (chair), Guy Blelloch, Gary Miller, Seth Pettie (Michigan)
J. Derryberry Adaptive Binary Search Trees 1 / 45
The Search Problem
Membership-testing, dictionary, successor/predecessor, etc.
Sequence of queries σ1 · · ·σm
Assume each σj ∈ 1, . . . , n
J. Derryberry Adaptive Binary Search Trees 2 / 45
The Search Problem
Membership-testing, dictionary, successor/predecessor, etc.
Sequence of queries σ1 · · ·σm
Assume each σj ∈ 1, . . . , n
J. Derryberry Adaptive Binary Search Trees 2 / 45
The Search Problem
Membership-testing, dictionary, successor/predecessor, etc.
Sequence of queries σ1 · · ·σm
Assume each σj ∈ 1, . . . , n
J. Derryberry Adaptive Binary Search Trees 2 / 45
Outline
1 Introduction
2 Lower Bounds and Competitiveness
3 The Unified Bound and Splay Trees
4 Cache-Splay Trees
5 Conclusion
J. Derryberry Adaptive Binary Search Trees 3 / 45
Outline
1 Introduction
2 Lower Bounds and Competitiveness
3 The Unified Bound and Splay Trees
4 Cache-Splay Trees
5 Conclusion
J. Derryberry Adaptive Binary Search Trees 4 / 45
Which Computational Model?
Hashing (RAM): O(1)
(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)
(requires integers, no augmenting)
Comparison model: O(lg n)
(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)
(requires integers, no augmenting)
Comparison model: O(lg n)
(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)
(requires integers, no augmenting)
Comparison model: O(lg n)
(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)(requires integers, no augmenting)
Comparison model: O(lg n)
(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)(requires integers, no augmenting)
Comparison model: O(lg n)
(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)(requires integers, no augmenting)
Comparison model: O(lg n)(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)(requires integers, no augmenting)
Comparison model: O(lg n)(no augmenting)
BST model: O(lg n)
(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
Which Computational Model?
Hashing (RAM): O(1)(but no successor)
vEB/Fusion Trees (RAM): O(√
lg n)(requires integers, no augmenting)
Comparison model: O(lg n)(no augmenting)
BST model: O(lg n)(supports augmenting)
J. Derryberry Adaptive Binary Search Trees 5 / 45
BST Model [Wil89]
x
y
y
x
A B
C A
B C
rotate x
rotate y
Binary tree in symmetric order
Modifiable with BST rotations
Must rotate queried node to root
J. Derryberry Adaptive Binary Search Trees 6 / 45
BST Model [Wil89]
x
y
y
x
A B
C A
B C
rotate x
rotate y
Binary tree in symmetric order
Modifiable with BST rotations
Must rotate queried node to root
J. Derryberry Adaptive Binary Search Trees 6 / 45
BST Model [Wil89]
x
y
y
x
A B
C A
B C
rotate x
rotate y
Binary tree in symmetric order
Modifiable with BST rotations
Must rotate queried node to root
J. Derryberry Adaptive Binary Search Trees 6 / 45
Flexibility of the BST Model
Can easily do O(lg n) worst-case
Can adapt to sequences with patterns
To adapt, rotate nodes likely to be accessed to near the root
J. Derryberry Adaptive Binary Search Trees 7 / 45
Flexibility of the BST Model
Can easily do O(lg n) worst-case
Can adapt to sequences with patterns
To adapt, rotate nodes likely to be accessed to near the root
J. Derryberry Adaptive Binary Search Trees 7 / 45
Flexibility of the BST Model
Can easily do O(lg n) worst-case
Can adapt to sequences with patterns
To adapt, rotate nodes likely to be accessed to near the root
J. Derryberry Adaptive Binary Search Trees 7 / 45
Outline
1 Introduction
2 Lower Bounds and Competitiveness
3 The Unified Bound and Splay Trees
4 Cache-Splay Trees
5 Conclusion
J. Derryberry Adaptive Binary Search Trees 8 / 45
Why Prove a Lower Bound?
Help prove optimality
Invalidate the computational model
Understand the model/problem
J. Derryberry Adaptive Binary Search Trees 9 / 45
Why Prove a Lower Bound?
Help prove optimality
Invalidate the computational model
Understand the model/problem
J. Derryberry Adaptive Binary Search Trees 9 / 45
Why Prove a Lower Bound?
Help prove optimality
Invalidate the computational model
Understand the model/problem
J. Derryberry Adaptive Binary Search Trees 9 / 45
Lower Bound Basics
Online BST costs Ω(lg n)?
Offline BST costs Ω(lg n)?
Instance-specific bounds?
J. Derryberry Adaptive Binary Search Trees 10 / 45
Lower Bound Basics
Online BST costs Ω(lg n)?
Offline BST costs Ω(lg n)?
Instance-specific bounds?
J. Derryberry Adaptive Binary Search Trees 10 / 45
Lower Bound Basics
Online BST costs Ω(lg n)?
Offline BST costs Ω(lg n)?
Instance-specific bounds?
J. Derryberry Adaptive Binary Search Trees 10 / 45
Wilber’s First Bound [Wil89]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
J. Derryberry Adaptive Binary Search Trees 11 / 45
Wilber’s First Bound [Wil89]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
J. Derryberry Adaptive Binary Search Trees 11 / 45
Wilber’s First Bound [Wil89]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
J. Derryberry Adaptive Binary Search Trees 11 / 45
Wilber’s First Bound [Wil89]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
J. Derryberry Adaptive Binary Search Trees 11 / 45
Wilber’s First Bound [Wil89]
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
J. Derryberry Adaptive Binary Search Trees 11 / 45
Interleave Bound [DHIP04]
13
1410
11
12
151
2
3
4
5 7
6
8
9
J. Derryberry Adaptive Binary Search Trees 12 / 45
Interleave Bound [DHIP04]
13
1410
11
12
151
2
3
4
5 7
6
8
9
J. Derryberry Adaptive Binary Search Trees 12 / 45
Interleave Bound [DHIP04]
13
1410
11
12
151
2
3
4
5 7
6
8
9
J. Derryberry Adaptive Binary Search Trees 12 / 45
Interleave Bound [DHIP04]
13
1410
11
12
151
2
3
4
5 7
6
8
9
J. Derryberry Adaptive Binary Search Trees 12 / 45
Interleave Bound [DHIP04]
13
1410
11
12
151
2
3
4
5 7
6
8
9
J. Derryberry Adaptive Binary Search Trees 12 / 45
Uses of the Interleave Bound
13
1410
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
1410
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
1410
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
14
10
11
12
15
1
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
Uses of the Interleave Bound
13
1410
11
12
151
2
3
4
5 7
6
8
9
Implications
Balanced BSTs dynamically optimal for random sequences
Rotate paths (only in BST) into a red-black tree [DHIP04]
Swap red-black trees for splay trees and gain [WDS06]
Can also rotate lower bound tree to allow updates [WDS06]
J. Derryberry Adaptive Binary Search Trees 13 / 45
How good is Wilber-1/Interleave?
Lets us compete (O(lg lg n)-competitiveness)
Every static lower bound tree is loose by Ω(lg lg n)
Can the dynamic interleave bound help?
Can Wilber’s second bound help?
J. Derryberry Adaptive Binary Search Trees 14 / 45
How good is Wilber-1/Interleave?
Lets us compete (O(lg lg n)-competitiveness)
Every static lower bound tree is loose by Ω(lg lg n)
Can the dynamic interleave bound help?
Can Wilber’s second bound help?
J. Derryberry Adaptive Binary Search Trees 14 / 45
How good is Wilber-1/Interleave?
Lets us compete (O(lg lg n)-competitiveness)
Every static lower bound tree is loose by Ω(lg lg n)
Can the dynamic interleave bound help?
Can Wilber’s second bound help?
J. Derryberry Adaptive Binary Search Trees 14 / 45
How good is Wilber-1/Interleave?
Lets us compete (O(lg lg n)-competitiveness)
Every static lower bound tree is loose by Ω(lg lg n)
Can the dynamic interleave bound help?
Can Wilber’s second bound help?
J. Derryberry Adaptive Binary Search Trees 14 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
MIBS: Generalizing Wilber-1 and Wilber-2 [DSW05]
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 15 / 45
The MIBS Bound
Theorem
The number of boxes is a lower bound on OPT(σ).
J. Derryberry Adaptive Binary Search Trees 16 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
associated rotation
J. Derryberry Adaptive Binary Search Trees 17 / 45
Proof of MIBS Bound
keyspace
tim
e1
23
45
67
89
10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1213
1415
11
associated rotation
J. Derryberry Adaptive Binary Search Trees 17 / 45
MIBS ≥ Wilber’s First Bound
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
keyspaceti
me
12
34
56
78
910
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1612
1314
1511
J. Derryberry Adaptive Binary Search Trees 18 / 45
MIBS ≥ Wilber’s First Bound
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
keyspaceti
me
12
34
56
78
910
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1612
1314
1511
J. Derryberry Adaptive Binary Search Trees 18 / 45
MIBS ≥ Wilber’s First Bound
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
keyspaceti
me
12
34
56
78
910
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1612
1314
1511
J. Derryberry Adaptive Binary Search Trees 18 / 45
MIBS ≥ Wilber’s First Bound
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
keyspaceti
me
12
34
56
78
910
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1612
1314
1511
J. Derryberry Adaptive Binary Search Trees 18 / 45
MIBS ≥ Wilber’s First Bound
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
keyspaceti
me
12
34
56
78
910
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1612
1314
1511
J. Derryberry Adaptive Binary Search Trees 18 / 45
Musing on MIBS
Generalizes previous bounds
Does not help improve competitiveness so far
Strengthens connection between partial-sums problem andBST model
J. Derryberry Adaptive Binary Search Trees 19 / 45
Musing on MIBS
Generalizes previous bounds
Does not help improve competitiveness so far
Strengthens connection between partial-sums problem andBST model
J. Derryberry Adaptive Binary Search Trees 19 / 45
Musing on MIBS
Generalizes previous bounds
Does not help improve competitiveness so far
Strengthens connection between partial-sums problem andBST model
J. Derryberry Adaptive Binary Search Trees 19 / 45
Partial-Sums Problem
Maintain an array under update operations
Return sum(1, . . . , i) when requested for current array values
Input: sequence of update and sum operations
Output: sequence of sums
Lower bound: Ω(lg n) in cell-probe model [PD04]
Lower bound: MIBS, if we require explicitly computed sums
J. Derryberry Adaptive Binary Search Trees 20 / 45
Partial-Sums Problem
Maintain an array under update operations
Return sum(1, . . . , i) when requested for current array values
Input: sequence of update and sum operations
Output: sequence of sums
Lower bound: Ω(lg n) in cell-probe model [PD04]
Lower bound: MIBS, if we require explicitly computed sums
J. Derryberry Adaptive Binary Search Trees 20 / 45
Partial-Sums Problem
Maintain an array under update operations
Return sum(1, . . . , i) when requested for current array values
Input: sequence of update and sum operations
Output: sequence of sums
Lower bound: Ω(lg n) in cell-probe model [PD04]
Lower bound: MIBS, if we require explicitly computed sums
J. Derryberry Adaptive Binary Search Trees 20 / 45
Partial-Sums Problem
Maintain an array under update operations
Return sum(1, . . . , i) when requested for current array values
Input: sequence of update and sum operations
Output: sequence of sums
Lower bound: Ω(lg n) in cell-probe model [PD04]
Lower bound: MIBS, if we require explicitly computed sums
J. Derryberry Adaptive Binary Search Trees 20 / 45
Partial-Sums Problem
Maintain an array under update operations
Return sum(1, . . . , i) when requested for current array values
Input: sequence of update and sum operations
Output: sequence of sums
Lower bound: Ω(lg n) in cell-probe model [PD04]
Lower bound: MIBS, if we require explicitly computed sums
J. Derryberry Adaptive Binary Search Trees 20 / 45
Partial-Sums Problem
Maintain an array under update operations
Return sum(1, . . . , i) when requested for current array values
Input: sequence of update and sum operations
Output: sequence of sums
Lower bound: Ω(lg n) in cell-probe model [PD04]
Lower bound: MIBS, if we require explicitly computed sums
J. Derryberry Adaptive Binary Search Trees 20 / 45
History of BST Lower Bounds
[Wil89]: first offline, instance-specific bounds
[BCK02]: offline information-theoretic bound
[DHIP04]: BST-like bound
[WDS06]: rotatable BST-like bound
[DSW05],[DHI+09]: generalization of Wilber-1 and Wilber-2
J. Derryberry Adaptive Binary Search Trees 21 / 45
History of BST Lower Bounds
[Wil89]: first offline, instance-specific bounds
[BCK02]: offline information-theoretic bound
[DHIP04]: BST-like bound
[WDS06]: rotatable BST-like bound
[DSW05],[DHI+09]: generalization of Wilber-1 and Wilber-2
J. Derryberry Adaptive Binary Search Trees 21 / 45
History of BST Lower Bounds
[Wil89]: first offline, instance-specific bounds
[BCK02]: offline information-theoretic bound
[DHIP04]: BST-like bound
[WDS06]: rotatable BST-like bound
[DSW05],[DHI+09]: generalization of Wilber-1 and Wilber-2
J. Derryberry Adaptive Binary Search Trees 21 / 45
History of BST Lower Bounds
[Wil89]: first offline, instance-specific bounds
[BCK02]: offline information-theoretic bound
[DHIP04]: BST-like bound
[WDS06]: rotatable BST-like bound
[DSW05],[DHI+09]: generalization of Wilber-1 and Wilber-2
J. Derryberry Adaptive Binary Search Trees 21 / 45
History of BST Lower Bounds
[Wil89]: first offline, instance-specific bounds
[BCK02]: offline information-theoretic bound
[DHIP04]: BST-like bound
[WDS06]: rotatable BST-like bound
[DSW05],[DHI+09]: generalization of Wilber-1 and Wilber-2
J. Derryberry Adaptive Binary Search Trees 21 / 45
History of BST Competitiveness
[BCK02]: dynamic search optimality
[DHIP04]: O(lg lg n)-competitive
[WDS06, DSW09]: support insert/delete, deque, working set
[Geo08]: like multi-splay, only O(lg lg n)-competitive
[BDDF09]: support worst-case O(lg n) running time
J. Derryberry Adaptive Binary Search Trees 22 / 45
History of BST Competitiveness
[BCK02]: dynamic search optimality
[DHIP04]: O(lg lg n)-competitive
[WDS06, DSW09]: support insert/delete, deque, working set
[Geo08]: like multi-splay, only O(lg lg n)-competitive
[BDDF09]: support worst-case O(lg n) running time
J. Derryberry Adaptive Binary Search Trees 22 / 45
History of BST Competitiveness
[BCK02]: dynamic search optimality
[DHIP04]: O(lg lg n)-competitive
[WDS06, DSW09]: support insert/delete, deque, working set
[Geo08]: like multi-splay, only O(lg lg n)-competitive
[BDDF09]: support worst-case O(lg n) running time
J. Derryberry Adaptive Binary Search Trees 22 / 45
History of BST Competitiveness
[BCK02]: dynamic search optimality
[DHIP04]: O(lg lg n)-competitive
[WDS06, DSW09]: support insert/delete, deque, working set
[Geo08]: like multi-splay, only O(lg lg n)-competitive
[BDDF09]: support worst-case O(lg n) running time
J. Derryberry Adaptive Binary Search Trees 22 / 45
History of BST Competitiveness
[BCK02]: dynamic search optimality
[DHIP04]: O(lg lg n)-competitive
[WDS06, DSW09]: support insert/delete, deque, working set
[Geo08]: like multi-splay, only O(lg lg n)-competitive
[BDDF09]: support worst-case O(lg n) running time
J. Derryberry Adaptive Binary Search Trees 22 / 45
Outline
1 Introduction
2 Lower Bounds and Competitiveness
3 The Unified Bound and Splay Trees
4 Cache-Splay Trees
5 Conclusion
J. Derryberry Adaptive Binary Search Trees 23 / 45
Formulaic Adaptivity
Strength of competitiveness depends on the model
Alternative: input-sensitive bounds with intuitive meaning
J. Derryberry Adaptive Binary Search Trees 24 / 45
Formulaic Adaptivity
Strength of competitiveness depends on the model
Alternative: input-sensitive bounds with intuitive meaning
J. Derryberry Adaptive Binary Search Trees 24 / 45
Bounds with Intuitive Meaning
Working Set Bound (exploiting temporal locality)
Access x , then t distinct keys. Pay O(lg t) to access x again.
Dynamic Finger Bound (exploiting spatial locality)
Access x . Then, access y at cost O(lg |x − y |).
J. Derryberry Adaptive Binary Search Trees 25 / 45
Bounds with Intuitive Meaning
Working Set Bound (exploiting temporal locality)
Access x , then t distinct keys. Pay O(lg t) to access x again.
Dynamic Finger Bound (exploiting spatial locality)
Access x . Then, access y at cost O(lg |x − y |).
J. Derryberry Adaptive Binary Search Trees 25 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
What If There Is a Mix of Spatial and Temporal Locality?
Need two fingers
1, n2 + 1, 2, n
2 + 2, 3, n2 + 3, . . .
Need three fingers
1, n3 + 1, 2n
3 + 1, 2, n3 + 2, . . .
Support many fingers: Unified Bound [Iac01]
Cost of a query to y :
minx
(lg tx + lg |x − y |)
J. Derryberry Adaptive Binary Search Trees 26 / 45
Relationship of Unified Bound to Dynamic Optimality
Generalizes working set and dynamic finger bounds
Iacono achived the bound in the pointer model
Not sufficient for optimality:
1, n1/2 + 1, 2n1/2 + 1, 3n1/2 + 1, . . . , (n1/2 − 1)n1/2 + 1,
2, n1/2 + 2, 2n1/2 + 2, 3n1/2 + 2, . . . , (n1/2 − 1)n1/2 + 2,
3, n1/2 + 3, 2n1/2 + 3, 3n1/2 + 3, . . . , (n1/2 − 1)n1/2 + 3,
...
n1/2, n1/2 + n1/2, 2n1/2 + n1/2, 3n1/2 + n1/2, . . . , n.
Necessary for optimality? Open until this work.
J. Derryberry Adaptive Binary Search Trees 27 / 45
Relationship of Unified Bound to Dynamic Optimality
Generalizes working set and dynamic finger bounds
Iacono achived the bound in the pointer model
Not sufficient for optimality:
1, n1/2 + 1, 2n1/2 + 1, 3n1/2 + 1, . . . , (n1/2 − 1)n1/2 + 1,
2, n1/2 + 2, 2n1/2 + 2, 3n1/2 + 2, . . . , (n1/2 − 1)n1/2 + 2,
3, n1/2 + 3, 2n1/2 + 3, 3n1/2 + 3, . . . , (n1/2 − 1)n1/2 + 3,
...
n1/2, n1/2 + n1/2, 2n1/2 + n1/2, 3n1/2 + n1/2, . . . , n.
Necessary for optimality? Open until this work.
J. Derryberry Adaptive Binary Search Trees 27 / 45
Relationship of Unified Bound to Dynamic Optimality
Generalizes working set and dynamic finger bounds
Iacono achived the bound in the pointer model
Not sufficient for optimality:
1, n1/2 + 1, 2n1/2 + 1, 3n1/2 + 1, . . . , (n1/2 − 1)n1/2 + 1,
2, n1/2 + 2, 2n1/2 + 2, 3n1/2 + 2, . . . , (n1/2 − 1)n1/2 + 2,
3, n1/2 + 3, 2n1/2 + 3, 3n1/2 + 3, . . . , (n1/2 − 1)n1/2 + 3,
...
n1/2, n1/2 + n1/2, 2n1/2 + n1/2, 3n1/2 + n1/2, . . . , n.
Necessary for optimality? Open until this work.
J. Derryberry Adaptive Binary Search Trees 27 / 45
Relationship of Unified Bound to Dynamic Optimality
Generalizes working set and dynamic finger bounds
Iacono achived the bound in the pointer model
Not sufficient for optimality:
1, n1/2 + 1, 2n1/2 + 1, 3n1/2 + 1, . . . , (n1/2 − 1)n1/2 + 1,
2, n1/2 + 2, 2n1/2 + 2, 3n1/2 + 2, . . . , (n1/2 − 1)n1/2 + 2,
3, n1/2 + 3, 2n1/2 + 3, 3n1/2 + 3, . . . , (n1/2 − 1)n1/2 + 3,
...
n1/2, n1/2 + n1/2, 2n1/2 + n1/2, 3n1/2 + n1/2, . . . , n.
Necessary for optimality? Open until this work.
J. Derryberry Adaptive Binary Search Trees 27 / 45
Why Should Splay Trees Satisfy the Unified Bound?
J. Derryberry Adaptive Binary Search Trees 28 / 45
Why Should Splay Trees Satisfy the Unified Bound?
J. Derryberry Adaptive Binary Search Trees 28 / 45
Why Is This Hard to Prove?
Unified Bound subsumes dynamic finger bound
Proof of dynamic finger bound is 80 pages! [CMSS00, Col00]
Also, Unified Bound subsumes the deque bound
Not proven for splay trees (best is α∗(n) [Pet08])
J. Derryberry Adaptive Binary Search Trees 29 / 45
Why Is This Hard to Prove?
Unified Bound subsumes dynamic finger bound
Proof of dynamic finger bound is 80 pages! [CMSS00, Col00]
Also, Unified Bound subsumes the deque bound
Not proven for splay trees (best is α∗(n) [Pet08])
J. Derryberry Adaptive Binary Search Trees 29 / 45
Why Is This Hard to Prove?
Unified Bound subsumes dynamic finger bound
Proof of dynamic finger bound is 80 pages! [CMSS00, Col00]
Also, Unified Bound subsumes the deque bound
Not proven for splay trees (best is α∗(n) [Pet08])
J. Derryberry Adaptive Binary Search Trees 29 / 45
Why Is This Hard to Prove?
Unified Bound subsumes dynamic finger bound
Proof of dynamic finger bound is 80 pages! [CMSS00, Col00]
Also, Unified Bound subsumes the deque bound
Not proven for splay trees (best is α∗(n) [Pet08])
J. Derryberry Adaptive Binary Search Trees 29 / 45
Skip-Splay: Adding a Small Amount of Structure
T1 T2 T√n
√n elements
T ′1 T ′
2 T ′n
14
n1/4 elements
J. Derryberry Adaptive Binary Search Trees 30 / 45
Skip-Splay: Adding a Small Amount of Structure
J. Derryberry Adaptive Binary Search Trees 30 / 45
Skip-Splay: Adding a Small Amount of Structure
J. Derryberry Adaptive Binary Search Trees 30 / 45
Simple BSTs Versus Simple Proofs
Pro
ofle
ngt
h
Splay
Splay requires 80-page proof for dynamic finger
Skip-splay requires 8-page proof for Unified Bound plus lg lg n
Cache-splay requires 4-page proof for Unified Bound
J. Derryberry Adaptive Binary Search Trees 31 / 45
Simple BSTs Versus Simple Proofs
Pro
ofle
ngt
h
Splay Skip-splay
Splay requires 80-page proof for dynamic finger
Skip-splay requires 8-page proof for Unified Bound plus lg lg n
Cache-splay requires 4-page proof for Unified Bound
J. Derryberry Adaptive Binary Search Trees 31 / 45
Simple BSTs Versus Simple Proofs
Pro
ofle
ngt
h
Splay Skip-splay Cache-splay
Splay requires 80-page proof for dynamic finger
Skip-splay requires 8-page proof for Unified Bound plus lg lg n
Cache-splay requires 4-page proof for Unified Bound
J. Derryberry Adaptive Binary Search Trees 31 / 45
Outline
1 Introduction
2 Lower Bounds and Competitiveness
3 The Unified Bound and Splay Trees
4 Cache-Splay Trees
5 Conclusion
J. Derryberry Adaptive Binary Search Trees 32 / 45
The Cache-Splay Hierarchy of Keys
Level-1 block contains b1 = 4 keys
Level-2 block contains b1 level-1 blocks (16 keys)
Level-i block contains bi−1 level-(i − 1) blocks (b2i−1 keys)
Level-lg lg n block contains all keys
J. Derryberry Adaptive Binary Search Trees 33 / 45
The Cache-Splay Hierarchy of Keys
Level-1 block contains b1 = 4 keys
Level-2 block contains b1 level-1 blocks (16 keys)
Level-i block contains bi−1 level-(i − 1) blocks (b2i−1 keys)
Level-lg lg n block contains all keys
J. Derryberry Adaptive Binary Search Trees 33 / 45
The Cache-Splay Hierarchy of Keys
Level-1 block contains b1 = 4 keys
Level-2 block contains b1 level-1 blocks (16 keys)
Level-i block contains bi−1 level-(i − 1) blocks (b2i−1 keys)
Level-lg lg n block contains all keys
J. Derryberry Adaptive Binary Search Trees 33 / 45
The Cache-Splay Hierarchy of Keys
Level-1 block contains b1 = 4 keys
Level-2 block contains b1 level-1 blocks (16 keys)
Level-i block contains bi−1 level-(i − 1) blocks (b2i−1 keys)
Level-lg lg n block contains all keys
J. Derryberry Adaptive Binary Search Trees 33 / 45
The Cache View and the Tree View
level 1 blocks
level 2 blocks
level 3 blocks
level 4 blocks
level 1 of T
level 2 of T
level 3 of T
level 4 of T
J. Derryberry Adaptive Binary Search Trees 34 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
The Cache View of a Query
x
Cache view of a query to x
Cache loop, iteration 1
Cache loop, iteration 2
Eject loop, iteration 1
Eject loop, iteration 2
Important Fact
If t keys have been queriedsince x , then the size of x ’scurrent block size is tO(1).
J. Derryberry Adaptive Binary Search Trees 35 / 45
BST Implementation of the Cache Loop
v z
w x y
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
w x y
v z
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x y
v zw
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x y
v zw
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
v zw
y
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
v zw
y
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
wy
v
z
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
wy
v
z
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
wy
vz
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
wy
vz
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
wy
vz
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
x
wy
vz
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
w
x
y
vz
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Cache Loop
vw y
zx
One cache loop iteration
splay(w)
splay(y)
splay(v)
splay(z)
incRoot(leftChild(w))
incRoot(rightChild(y))
decRoot(w)
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 36 / 45
BST Implementation of the Eject Loop
vw y
z
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
vw y
z
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
yz
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
yz
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v
y
z
w
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
BST Implementation of the Eject Loop
v z
w y
One eject loop iteration
splay(v)
splay(z)
splay(w)
splay(y)
incRoot(w)
decRoot(leftChild(w))
decRoot(rightChild(y))
Each operation costs O(lg(block size for lower level))
J. Derryberry Adaptive Binary Search Trees 37 / 45
The Cost of a Query
size of this block is B
x
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B4
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B4
lg B4
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B4
lg B4
12 lg B
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B4
lg B4
12 lg B
lg B
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B4
lg B4
12 lg B
lg B
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
The Cost of a Query
size of this block is B
lg B
12 lg B
lg B4
lg B4
12 lg B
lg B
lg B + 12 lg B + 1
4 lg B + 14 lg B + 1
2 lg B + lg B = O(lg B)
Lemma (Query Cost)
A query to level i costs O(lg bi ) = O(lg(time since queried)).
J. Derryberry Adaptive Binary Search Trees 38 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
x y
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
x y
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Adding an Offset to the Blocks of the Cache
x y
Lemma (Offset Query Cost)
A query to level i of the “virtual cache” costs amortized O(lg bi ),which is O(lg(time since virtual block queried)).
Proof.
Potential of lg bj for each level-j block that is overlapped by alevel-j virtual block that is at a higher level in the virtual cache.
J. Derryberry Adaptive Binary Search Trees 39 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi ,
then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi ,
then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Randomizing the Offset
x y
Lemma (Random Offset)
For random offset, are x and y in different level-i virtual blocks?
If |x − y | ≥ bi , then probability = 1.
If |x − y | < bi , then probability = |x−y |bi
.
J. Derryberry Adaptive Binary Search Trees 40 / 45
Cache-Splay Satisfies the Unified Bound
x
y
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is expected O(
minx(
lg tx + lg |x − y |
)
).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Cache-Splay Satisfies the Unified Bound
x
y
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is expected O(
minx(
lg tx + lg |x − y |
)
).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Cache-Splay Satisfies the Unified Bound
x
y
O(lg t)
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is expected O(
minx(
lg tx + lg |x − y |
)
).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Cache-Splay Satisfies the Unified Bound
x
y
O(lg |x− y|)
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is expected O(
minx(
lg tx + lg |x − y |
)
).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Cache-Splay Satisfies the Unified Bound
x
y
O(lg |x− y|)
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is expected O(
minx(
lg tx + lg |x − y |
)
).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Cache-Splay Satisfies the Unified Bound
x
y
O(lg |x− y|)
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is expected O( minx( lg tx + lg |x − y |)).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Cache-Splay Satisfies the Unified Bound
x
y
O(lg |x− y|)
Proof.
Suppose we query x , then tx other keys, then y .
Then x ’s level in virtual cache has block size bi = tO(1)x .
If |x − y | < bi , then cost is expected O(lg tx).
If |x − y | ≥ bi , then cost is expected O(lg |x − y |).Cost is ////////////expected O( minx( lg tx + lg |x − y |)).
J. Derryberry Adaptive Binary Search Trees 41 / 45
Implications
Splay trees must satisfy the Unified Bound to beO(1)-competitive
Search for even more general formulaic bounds?
J. Derryberry Adaptive Binary Search Trees 42 / 45
Implications
Splay trees must satisfy the Unified Bound to beO(1)-competitive
Search for even more general formulaic bounds?
J. Derryberry Adaptive Binary Search Trees 42 / 45
Outline
1 Introduction
2 Lower Bounds and Competitiveness
3 The Unified Bound and Splay Trees
4 Cache-Splay Trees
5 Conclusion
J. Derryberry Adaptive Binary Search Trees 43 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Contributions
Lower boundsDynamic interleave bound [WDS06]MIBS bound [DSW05]: generalization of Wilber’s bounds thatis computable in polynomial time...also shows the strength of the BST model because it is alower bound for partial-sums (in thesis)
BST upper boundsMulti-splay [WDS06, DSW09]: first O(lg lg n)-competitiveBST to achieve other properties of an optimal BSTSkip-splay [DS09]: simple BST similar to splaying withinadditive O(lg lg n) of the Unified BoundCache-splay (in thesis): more complicated splay-basedalgorithm that achieves the Unified Bound (open since [Iac01])
Other resultsHigh-dimensional finger search [DSSW08]: first finger searchdata structure for k-d approximate nearest-neighborExperiments with bipartite parametric max-flow [BDG+07]Easy instances of combinatorial auctions [CDS04]
J. Derryberry Adaptive Binary Search Trees 44 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
Future Work
Better bounds for splaying: Unified Bound,o(lg n)-competitiveness, digit-reversal permutation,generalization of the Unified Bound, working set for splayingwithout rotate-to-root, new toolbox for analyzing splay treeswith splaying over induced subtrees.
Better bounds for any BST: use better lower bounds to showo(lg lg n)-competitiveness for some BST, show that someformulaic bound that implies BST competitiveness to within ao(lg n) factor
Further justification for the BST model itself: show MIBS is alower bound for partial-sums in a more general model, reduceBST model to partial-sums problem.
Thanks!
J. Derryberry Adaptive Binary Search Trees 45 / 45
[BCK02] Avrim Blum, Shuchi Chawla, and Adam Kalai. Static optimalityand dynamic search-optimality in lists and trees. In Proceedings ofthe 13th ACM-SIAM Symposium on Discrete Algorithms (SODA2002), pages 1–8, Philadelphia, PA, USA, 2002. Society forIndustrial and Applied Mathematics.
[BDDF09] Prosenjit Bose, Karim Douıeb, Vida Dujmovic, and Rolf Fagerberg.An o(log log n)-competitive binary search tree with optimalworst-case access times. Obtained on December 7, 2009 from:http://cgm.cs.mcgill.ca/ vida/pubs/papers/ZipperTrees.pdf, 2009.
[BDG+07] Maxim Babenko, Jonathan Derryberry, Andrew Goldberg, RobertTarjan, and Yunhong Zhou. Experimental evaluation of parametricmax-flow algorithms. In Proceedings of the 6th Workshop onExperimental Algorithms (WEA 2007), pages 256–269, 2007.
[CDS04] Vincent Conitzer, Jonathan Derryberry, and Tuomas Sandholm.Combinatorial auctions with structured item graphs. In Proceedingsof the 19th National Conference on Artificial Intelligence (AAAI2004), pages 212–218. AAAI Press / The MIT Press, 2004.
[CMSS00] Richard Cole, Bud Mishra, Jeanette Schmidt, and Alan Siegel. Onthe dynamic finger conjecture for splay trees. part I: Splay sortinglog n-block sequences. SIAM Journal on Computing, 30(1):1–43,2000.
[Col00] Richard Cole. On the dynamic finger conjecture for splay trees. partII: The proof. SIAM Journal on Computing, 30(1):44–85, 2000.
J. Derryberry Adaptive Binary Search Trees 45 / 45
[DHI+09] Erik D. Demaine, Dion Harmon, John Iacono, Daniel Kane, andMihai Patrascu. The geometry of binary search trees. InProceedings of the 20th ACM-SIAM Symposium on DiscreteAlgorithms (SODA 2009), pages 496–505, 2009.
[DHIP04] Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Patrascu.Dynamic optimality – almost. In Proceedings of the 45th IEEESymposium on Foundations of Computer Science (FOCS 2004),pages 484–490, 2004.
[DS09] Jonathan C. Derryberry and Daniel D. Sleator. Skip-splay: Towardachieving the unified bound in the bst model. In Proceedings of the11th International Symposium on Algorithms and Data Structures(WADS 2009), pages 194–205, Berlin, Heidelberg, 2009.Springer-Verlag.
[DSSW08] Jonathan Derryberry, Don Sheehy, Daniel D. Sleator, and MaverickWoo. Achieving spatial adaptivity while finding approximate nearestneighbors. In Proceedings of the 20th Canadian Conference onComputational Geometry (CCCG 2008), pages 163–166, 2008.
[DSW05] Jonathan Derryberry, Daniel Dominic Sleator, and Chengwen ChrisWang. A lower bound framework for binary search trees withrotations. Technical Report CMU-CS-05-187, Carnegie MellonUniversity, 2005.
J. Derryberry Adaptive Binary Search Trees 45 / 45
[DSW09] Jonathan Derryberry, Daniel Sleator, and Chengwen Chris Wang.Properties of multi-splay trees. Technical Report CMU-CS-09-171,Carnegie Mellon University, 2009.
[Geo08] George F. Georgakopoulos. Chain-splay trees, or, how to achieveand prove log log n-competitiveness by splaying. InformationProcessing Letters, 106(1):37–43, 2008.
[Iac01] John Iacono. Alternatives to splay trees with o(log n) worst-caseaccess times. In Proceedings of the 12th ACM-SIAM Symposiumon Discrete Algorithms (SODA 2001), pages 516–522, Philadelphia,PA, USA, 2001. Society for Industrial and Applied Mathematics.
[PD04] Mihai Patrascu and Erik D. Demaine. Tight bounds for thepartial-sums problem. In In Proceedings of the 15th ACM-SIAMSymposium on Discrete Algorithms (SODA 2004), pages 20–29,Philadelphia, PA, USA, 2004. Society for Industrial and AppliedMathematics.
[Pet08] Seth Pettie. Splay trees, davenport-schinzel sequences, and thedeque conjecture. In Proceedings of the 19th ACM-SIAMSymposium on Discrete Algorithms (SODA 2008), pages1115–1124, 2008.
[WDS06] Chengwen Chris Wang, Jonathan Derryberry, and Daniel DominicSleator. O(log log n)-competitive dynamic binary search trees. InProceedings of the 17th ACM-SIAM Symposium on Discrete
J. Derryberry Adaptive Binary Search Trees 45 / 45
Algorithms (SODA 2006), pages 374–383, New York, NY, USA,2006. ACM.
[Wil89] Robert Wilber. Lower bounds for accessing binary search trees withrotations. SIAM Journal on Computing, 18(1):56–67, 1989.
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